UC-NRLF 


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8061  'iJ'NVriVd 


On  the  In-and-Circumscribed 
Triangles  of  the  Plane 
Rational  Quartic  Curve 


By 
Joseph  Nelson  Rice 


A  DISSERTATION 

Submitted  to  the  Faculty  of  Sciences  of  the  Catholic  University  of 

America  in  partial  fulfilment  of  the  requirements  for 

the  degree  of  Doctor  of  Philosophy. 


Washington,  D.  C. 
June,  1917. 


On  the  In-and-Circumscribed 
Triangles  of  the  Plane 
Rational  Quartic  Curve 


By 
Joseph  Nelson  Rice 


A  DISSERTATION 

Submitted  to  the  Faculty  of  Sciences  of  the  Catholic  University  of 

America  in  partial  fulfilment  of  the  requirements  for 

the  degree  of  Doctor  of  Philosophy. 


.•      .    .    .  :v:'\*;ri'. 


Washington,  D.  C. 
June,  1917. 


•-  ;  i^-"  .' 


NATIONAL  CAPITAL  PRESS,   INC.,  WASHINGTON,   0.  C. 


ON  THE  IN-AND-CIRCUMSCRIBED  TRIAN- 
GLES OF  THE  PLANE  RATIONAL  QUARTIC 
CURVE 

INTRODUCTIOX 

The  question  of  the  existence  of  simultaneously  inscribed  and 
circumscribed  triangles  has  received  considerable  attention,  most 
of  which,  however,  has  been  directed  to  the  consideration  of  poristic 
cases. 

R.  \.  Roberts^  investigates  the  possibility  of  the  existence  of 
an  infinite  number  of  closed  polygons  simultaneously  inscribed 
in,  and  circumscribed  about,  a  unicursal  quartic.  In  this  connec- 
tion he  discusses  only  the  case  of  the  nodo-bicuspidal  quartic, 
and  shows  that,  for  the  triangle,  the  results  are  irrelevant. 

Morley,"  in  his  article  entitled  "The  Poncelet  Polygons  of  a 
Limacon,"  shows  that  the  results  obtained  are  poristic,  except 
for  the  triangle,  in  which  case  they  are  irrelevant. 

Cayley,^  in  "On  a  Triangle  In-and-circumscribed  about  a  Quartic 
Curve,"  shows  that  the  binodal  quartic 


('-;)■ 


has  four  such  triangles.     These  are  such  that  two  of  the  sides  are 
tangent  to  the  inner  loop  and  the  third  to  the  outer. 

The  same  author,  in  "On  the  Problem  of  the  In-and-circum- 
scribed Triangle,"  considers  the  following  problem:  "required 
the  number  of  triangles  the  angles  of  which  are  situate  in  a  given 
curve  or  curves,  and  the  sides  of  which  touch  a  given  curve  or 
curves."^  He  discusses  52  cases  of  the  problem,  according  as  the 
curves  containing  the  angles  or  touching  the  sides  are,  or  are  not, 
distinct  curves.     The  simplest  case  is  when  all  the  curves  are 

1  Proceedings  of  the  London  Mathematical  Society,  Vol.  xvi,  p.  53. 

2  Ibid.,  Vol.  xxix,  pp.  8,3-97. 

3  Collected  Mathematical  Works,  Vol.  v,  pp.  489-492. 

Philosophical  Transactions,  t.  XXX  (1865),  pp.  340-342. 

4  Collected  Mathematical  Works,  Vol.  viii,  pp.  212-258. 

Philosophical  Transactions,  t.  clxi  (1871),  pp.  369-412. 

5  Collected  Mathematical  Works,  Vol.  viii,  p.  567. 

Reports  of  the  British  Associations  for  the  Adrancement  of  Science,  1865-1873. 
Report,  1870,  pp.  9-10. 

3 


:Mj2L0i 


4         •Tiib-&iid-Circnnt8cribed  Triangles  of  Quurtic  Curve 

distinct;  the  number  of  triangles  is  here  equal  to  2aceBDF,  where 
a,  c,  e  are  the  orders  of  the  curves  containing  the  angles  respec- 
tively; and  B,  D,  F  are  the  classes  of  the  curves  touched  by  the 
sides  respectively.  The  last  and  most  difficult  case  is  when  the 
six  curves  are  all  of  them  one  and  the  same  carve.  The  number 
of  triangles  is  here  equal  to  one-sixth  of 

-46(.43+a3)-420(.42a+.4a2)  +  221(.42+a2)+704Ja 
+  172(.4+a)-fa[-9(^2_^a2)-12.4a+135(.4+a)-600] 

where  a  is  the  order,  .4  the  class  of  the  curve;  a  is  the  number, 
three  times  the  class +number  of  cusps,  or,  what  is  the  same,  three 
times  the  order + the  number  of  inflexions. 

It  is  to  be  noted  that  this  formula  gives  the  same  number  of 
triangles  as  has  been  found  by  the  method  used  later.  For 
example,  in  the  case  of  the  rational  quartic,  where  a  =  4,  .4  =  6, 
a  =  18,  the  number  of  triangles  is  8,  which  corresponds  to  that 
found  on  page  18.  For  the  cuspidal  quartic,  where  a  =  4,  A  — 5, 
a  =16,  the  number  is  two,  which  alsD  corresponds  to  the  number 
found  on  page  2*2. 

In  this  paper  it  is  proposed  to  look  into  the  existence  and  actual 
number  of  such  triangles  for  the  following  types  of  rational  quartics : 
I.  Quartic  with  three  double  points. 
II.  Quartic  w^ith  one  double  point  and  a  tacnode. 

III.  Quartic  with  a  triple  point. 

IV.  Quartic  with  two  double  points  and  a  cusp. 

This  discussion  was  led  up  to  by  preliminary  work  on  the  three- 
cusped  rational  quintic.  Upon  subjection  to  a  quadratic  trans- 
formation this  curve  goes  into  a  rational  quartic,  wdiich,  it  will 
be  show^n,  has  triangles  of  the  kind  here  mentioned.  Accordingly, 
it  will  first  be  proved  that  the  quintic  can  have  certain  conditions 
imposed  upon  its  coefficients  so  that  it  may  acquire  an  additional 
cusp  or  a  tacnode  without  degenerating.  It  will  also  be  shown 
that  it  cannot  have  a  triple  point. 


THE  THREE-CUSPED  RATIONAL  QUINTIC 

This  quintic  may  be  expressed  parametrically  as  follows: 

that  is,  x,=  (t-ai)(t-l32)Ht-l3s)' 

This  quintic  has  cusps  at  the  vertices  of  the  fundamental  tri- 
angle, the  values  of  the  parameters  thereat  being 

t^^i,     (i  =  l,  2,  3) 
as  is  readily  seen  by  considering  the  common  intersections  on 

any  two  of  the  lines 

.r,  =  0,     (I'-l,  2,  3) 

Consider,  for  example,  the  intersections  on  a;i  =  0  and  x^  =  0. 
They  have  in  common  /Ss  taken  twice.  These  common  points 
must  be  at  the  intersection  of  the  lines  themselves,  viz.,  at  the 
vertex  three. 

For  the  purpose  of  more  ready  computation,  specialize  this 
quintic  by  letting  the  values  of  the  parameters  at  the  cusps  be 
0,  1,  00.  The  parametric  equations  of  the  curve  may  then  be 
written  as : 


i-oc 

Xi  = 


r- 


Xi  =  (t-a)if-iy 


xz^t-y  x,=  {t-y){t-l)H^^ 

The  Plucker  numbers  of  the  curve  are: 

m  =  5,  5  =  3,  •    K  =  3 

n  =  5,  T  =  3,  t  =  3 

It  is  proposed  to  derive  the  conditions,  which  must  be  imposed 
upon  the  parameters  a,  (3,  y,  so  that  the  quintic  may  have, 

I.  A  fourth  cusp, 
II.  A  tacnode, 
III.  A  triple  point. 

It  will  be  seen  that,  excluding  the  case  of  the  triple  point,  the 
quintic  can  have  such  extra  points  in  addition  to  the  three  cusps 


6  In-and-Circumscrihed  Triangles  of  Qiiartic  Curve 

at  the  vertices    of    the  fundamental  triangle.     As  preliminary, 
there  will  be  derived : 

( 1)  The  cubic  equation  connecting  the  three  parameters  at  the 
three  flexes; 

(2)  The  sextic  equation  connecting  the  six  parameters  at  the 
three  double  points. 

(1)  The  Cubic  whose  Three  Roots  are  the  Flexes. 
Let  Lxi-{-Mx2-{-Nx3  =  0  be  a  flex-tangent. 

Then  Lfi{t)  +  J//2(/)  -\-Nf3(t)  =  0  has  a  triple  root, 

and  Z//(0  -\-Mfo\t)-\-Nf/(t)  =  0  has  the  same  root  doubled, 

also  Lfi'\i)  -\-]\W(i)  +yfs"{t)  =  0  has  the  same  root  once. 

From  these  three  equations  eliminate  L,  M,  N,  then 


Mt)     m     hit) 

//(/)       //(O       Uit) 

fl"(t)  f2"(t)  h"{t) 


0,  is  the  cubic  of  flexes. 


For  the  functions  of  /  and  their  derivatives  substitute  their  corre- 
sponding values  and  reduce,  thus  giving  as  the  required  equation: 

+  (7-/3-b2ai3-4a7-2^7)/  +  3a(i3-7  +  2^7)  =  0.  .  .  .  (1) 

If  one  of  these  flexes  coincides  with  one  of  the  three  cusps, 
that  is,  if  t  has  any  one  of  the  values  0,  1,  co,  there  is  then  at  this 
point  a  cusp  of  the  second  kind.*  Hence,  the  condition  that 
there  be  a  cusp  of  this  character  at : 

/-O     is  /3-7  +  !2/37  =  0 

/=!     .  .  3a+7-2a7-2  =  0 
/=  ex.   .  .  o:_^_|_2  =  0 

(2)  The  Sextic  connecting  the  Six  Parameters  at  the  Double 
Points. 

Let  X  and  ix  be  the  parameters  at  one  of  the  double  points. 

^-=^-(m-t)=^"-(X-7)   _ 


*  Salmon:  Higher  Plane  Curves  (French  Edition),  Chap,  ii,  p.  70. 
Cayley:  "On  the  Cusp  of  the  Second  Kind  or  Nodecusp;"  Collected  Mathe- 
matical WorJiK,  Vol.  V,  pp.  2fi.5.  266. 

Quarterly  Journal  of  Pure  and  Applied  Mathematics,  Vol.  vi  (1864),  pp.  74,  75. 


In-and-Circumscribed  Triangles  of  Quartic  Curve  7 

These  equations  reduce  to 

(\-a)fx^-\-{\-a)(X-y)pL-a\(\-y)  =  0 (1) 

(X-|8)M"'+(X-/3)(X-7-^2)M-(^X2-/37X-^2i3X  +  2j87+i3-7)  =  0  (2) 

Eliminate  /j,  from  these  equations.     The  eliminant  is 

I  (X-«)(X-7)      (X-/3)(X-7-2)  I 
I         (X-a)  (X-/3)  r 

I   (X-a)(X-7)  -aX(X-7)  1 

I  (\-l3){\-j-2)       -(^X2-/37X-2/3X+2j87+/3-7)  I 

I  -aX(X-7)      -(^X^-/37X-2i3X+2/37+/3-7)  |2_^ 
"^1       (X-a)  (X-^)  I     -" 

On  expanding  this  gives 

(a-i8)(a-/3  +  2)X6-2(a-/3)(a-^  +  2)(l  +  7)X^ 

+  {2(a-/3)(7-+a7-«i3  +  2a:+/3+37)-2(2^7  +  i3-7) 
+  («7-/37-2/3)2|  X-*     • 

-2!(a+^)(/37"'+«7-+a/3+2/37  +  3a7)-2(a+^+7) 

(2^7+^-7)-(«7-/37-2/3)(2^7  +  2ai3+^-7)j  X^ 

-{ 2(2/37+^-7)  (a2y_a/3-y,4-«2  +  2a7+i37)+2a/37(a-/3)  (7+2) 
-(2/37  +  2a^+/3-7)-iX- 

+  2a(2/37+i3-7)(7-/5)(«+l)X+a^(^-7)(2/37+/3-7)=0.  .  .  (3) 

It  may  be  noted  here  that,  according  to  Cayley's  analysis,*  a 
cusp  of  the  second  kind  takes  up  a  double  point.  If  a  — /S+2  =  0, 
then  the  sextic  has  /=  oo  for  a  double  root.  But  this  is  the  condi- 
tion that  the  quintic  have  a  node-cusp  at  /  =  <» . 

Equations  (1)  and  (2)  may  be  written  as  follows: 

(Tia2  —  cx(ai^  —  a2)~'yo'2~\~c(y<T2  —  0 (4) 

cTia2-/3((7i'^-<T2)-(7+2)(ro+(/37  +  2/3)(Ti-(/3-7  +  2i37)  =  0..  (5) 

where  ai=X+ju  and  o-2  =  Xju. 

Eliminating  a^  from  (4)  and  (o),  the  result  is: 

(a-i3)ai3-2(a-/3)(7+l)ai2+72(a-^)  +  2a(|3+7) 

-(/3-7  +  4/37)(ri-(a-7)(/3-7  +  2/37)=0 (6) 

/.   The  Condition  for  a  Fourth  Cusp 

It  has  been  seen  that  this  quintic  has  cusps  at  the  three  vertices 
of  the  fundamental  triangle,  the  values  of  the  parameters  at  these 
points  being  0,  1,  co.  If  a  fourth  cusp  be  possible,  let  the  value 
of  the  parameter  thereat  be  t  =  T. 

*Cayley:  "On  the  Cusp  of  the  Second  Kind  of  Xode-Cusp;"  Collected 
Mathematical  Worku,  Vol.  v,  pp.  265,  266. 


In-and-Circumscribed  Triangles  of  Quartic  Curve 


Join  Ir.     The  equation  giving  the  parameters  on  the  join  is 

r- 


T  —  a 

~^2 


{t-iy 
{r-iy 

0 

t  —  T 

Throwing  out  the  factor  -. -r-„, 

this  becomes: 


t-y 


T  —  7 
0 


0 


putting  t  —  T,  and  reducing, 


2r2-(3i3+7)r  +  (/3-7  +  2i37)  =0 (1) 

Similarly,  the  equation  giving  the  parameters  on  the  join  of  2t  is. 


t-a 

T—a 

0 


{t-\Y 

1 


/-7 

T  —  7 
0 


=  0 


which  reduces  to: 


(2) 


2 

-3/3-7 

0 

2 

2 

-3a-7 

0 

2 

=  0 


2T2-(3a  +  7)r  +  2a7  =  0 

Eliminate  r  from  (1)  and  (2).     The  eliminant  is, 

i3-7  +  2/37  0 

-3/3-7  /3-7  +  2^7 

2a7  0 

—  3a  —  7  2a7 

which  reduces  to: 

272(a-/3)2-(57-9a)(«-i3)(i3-7)+2(i3-7)-  =  0 

which  is  the  required  condition. 

A  cusp  may  result  by  the  coming  together  of  two  neighboring 
inflexions  as  is  readily  seen  by  considering  a  penultimate  case. 
Hence  the  condition  for  a  fourth  cusp  should  be  contained,  as  a 
factor,  in  the  discriminant  of  the  equation  of  the  flex  cubic 
(Equation  I,  p.  6). 

This  was  not  shown  in  general,  as  the  work  involved  considerable 
algebraic  difficulty.  In  the  special  cases  attempted,  it  was,  how- 
ever, proven  to  be  true,  as  the  following  example  will  show. 

As  previously  shown,  the  condition  for  a  cusp  of  the  second  kind 
at  /  =  00 ,  is : 

a-i3+2=0 

The  discriminant  of  the  flex-cubic  is  then  the  eliminant  of 

16(2a+ 1)«-  (2+37  +  25a  +  6a7-}-  12a")  =  0 
8(2a+ 1)^2 _  2(2+37  + 25a 4- f)a7+12a2)/  +  9a(2+a  +  37  +  2a7)  =  0 


I)i-and-Circi(niscrihed  Triangles  of  Quartic  Curve  9 

Therefore  E=  (2+3y-\-'25a-\-6ay-\-na^)^ 

-96a('2a-\-l)(2-\-a-\-3y-\-2ay) 

=  lUa*-'i^0a^y-\-36a^y^+40Sa^-396a'-y+3Qay^ 
-  193a2-  lUay-\-9y'i-^  Uy  +  4  =  0 

The  condition  for  a  fourth  cusp  reduces  to 

ia''-6ay-\-7a-3y-2  =  0 
This  divides  into  E,  giving  for  quotient, 

36a--Qay  +  39a-3y-2  =  0 

II.   The  Condition  for  a  Tacnode 

A  tacnode  may  arise  by  the  coming  together  of  two  double 
points.  The  joins  of  the  three  double  points,  two  and  two, 
intersect  the  quintic  in  an  additional  fifth  point  each.  If  two  of 
the  double  points  coincide,  then  two  of  these  additional  points 
will  also  coincide.  It  is  necessary  to  find  the  cubic  equation 
connecting  these  three  points. 

Let  AiXi-\-A2X2-{-A3X3 

=Ay(t-a)(t-l)-'+Ao(t-0)r+A3(t-y)(t-l)H-' 

=  Azt'-{2^y)A3t'+\As{l^2y)-hAo-^Ai\P-\A'y+Ao^ 

+  Ji(2+«)1  r~-\-Ai(l-{-2a)t-A,a  =  0, 

be  the  equation  of  the  five  parameters  of  the  points  of  intersection 
of  any  line  with  the  quintic. 

Then  si==2-\-y  is  the  sum  of  these  five  parameters, 

and  -  = is  the  sum  of  their  reciprocals. 

From  the  sextic  of  the  parameters  of  the  double  points,  it  is 
seen  that  5/=  2(1+7)  is  the  sum  of  the  six  parameters  thereat, 

and  --,  = is  the  sum  of  their  reciprocals. 

Se  a 

It  is  readily  seen  that,  if  Su  S2,  S3  be  the  symmetric  functions 
of  the  three  extra  points  of  intersection  with  the  curve  of  the 
joins  of  the  double  points,  two  and  two,  then, 

Si  =  35i-25/  =  3(2+7)-4(l+7)  =  2-7 

'^2     0^4     ^Sr,'     3  +  6a     4a+4      2a- 1 

and  Tr  =  3 2-7= = 

03       s^       Sq  a  a  a 


10         In-and-Circumscrihed  Triangles  of  Quartic  Curve 

To  determine  S3: 

Let  the  parameters  at  the  double  points  be  ti  and  <2,  ^3  and  ti, 
t;,  and  te.     Then  the  remaining  parameter  on  the 

join  of  (^1,  t2)  and  (ts,  U)  is  (2+7)  — (/i+/2+i3+i4) 
join  of  {h,  h)  and  {h,  U)  is  (2+7)-(^i+^2+^5+/6) 
join  of  (^3,  /4)  and  {h,  U)  is     (2+7)  -  (^3+/4+/5+/6) 

Si  =  3(2+7)-22<i  =  3(2+7) -4(1+7)  =2-7. 

Sz  =  {  (2  +  7)  -  (^1  +  ^2  +  ^3  +  ^4)  }  {  (2  +  7)  -  (/l  +  /2  +  /5  +  ^6)  }  {  (2  +  7) 

-{h-\-ti+h-\-te)} 
=  (2+7)'-(2+7)'(22/0  +  (2+7)[(S//)-  +  (/i+/2)(/3+/4) 

+  (/l  +  /2)(/5  +  /6)  +  (/3  +  /4)(/5  +  /6)l-2/.i(/i  +  /2)(/3  +  /4) 

+  (^l  +  ^2)(/5  +  /6)  +  (/3  +  ^4)(/5  +  ^6)}+(/l  +  /2)(/3  +  /4)(/5  +  /6) 

From  equation  (6),  page  7, 

s/.-= 2(7+1) 

(^l  +  4)(/3  +  ^4J  +  (/l  +  /2)(<5  +  ^6)+(/3  +  ^4)(/5  +  /6) 

^7^(«-/3)  +  2a(^+7)-(^-7  +  4i37) 

a  —  p 
Substitute  these  values  in  Ss  and  reduce: 

aW-y) 

'.^3  = ^ 

a  —  p 

But  |  =  ?^1 

03  a 

rj.,        f                                  ^       (2a-l)(/3-7) 
I  hereiore  02  = 7; • 

a  —  p 

The  cubic  equation  connecting  the  parameters  of  the  three 
required  points  of  intersection  is : 

a—p  a—p 

(a-i3)X'^-(a-i3)(2-7)X-  +  (2a-l)(|3-7)X-a(^-7)  =  0 

If  this  cubic  have  a  double  root,  it  means  that  two  of  the  double 
roots  coincide,  thus  giving  a  tacnode.  The  cubic  will  have  a 
double  root  if  its  discriminant  is  zero.  The  discriminant  is  the 
eliminant  of 

S(a-^)\2  +  2(a-i8)(7-2)X  +  (2a-l)(i3-7)=0 
(a-/3)f7-2)X2  +  2(2a-l)(/3-7)X-3a(,8-7)-0 


I)i-and-Circiiinsciibed  Tr'uuigles  of  Quartic  Curve 


11 


E  = 


3{a-^)        2(a-/3)(T-2)      (^2a-l)(l3-y)  0 

0  3(a-^)  2(a-/3)(7-2)    (2a-l){^-y) 

(«-/3)(t-2)  2(2a-l)(^-T)        -3a(^-7)  0 

0  («-|3)(7-2)     2(2a-l)(/3-7)     -3a(/3-7) 


=  3(a-|3)(i3-7)(a-T)[4a^T-(i3-a)+4«(8a^2^a7--3a^7-/372 
-5/3'-7)  +  (36a|37-61a2^+13a27+13ai32-a72+^72_^2^) 

-\-^{8a--5ay-Sa(3-l3y-{-^-)-\-My-^)]  =  0 

a  =  (3,  /3  =  7,  7=a  are  the  conditions  that  a  branch  of  the  curve 
passes  through  one  of  the  cusps.  Since  the  fifth  intersection  on 
the  sides  of  the  fundamental  triangle  is  given  by  t=a,  13,  or  y,  if 
two  of  these  are  equal,  then  the  points  of  intersection  must  be  at 
points  common  to  the  sides  of  the  triangle,  viz.,  the  vertices,  at 
which  are  the  cusps.  The  remaining  factor  then  must  be  the 
condition  that  the  quintic  acquire  a  tacnode. 

The  Triple  Point 

If  the  quintic  can  have  a  triple  point,  let  the  value  of  the  para- 
meter thereat  be  /  =  r. 

The  join  of  Ir  is 


t-a 
T  —  a 


1 


{t-\y 
0 


t-y 

T  —  7 
0 


=  0 


which  reduces  to 


The  join  of  2r  is 


(r-/3)/2  +  (x-^)(r-7-2)^  + 
■i3r^>/3(7+2)r- (^-7  +  2/37)  1=0. 
t-a 

T  —  a. 


(1) 


0 


{t-\r 

r-/3 

1 


t-y 

T  —  y 
0 


=  0 


which  reduces  to 

(T-a)^2^(r-7)(r-a)/-«r(r-7)=0 (2) 

These  two  equations  give  the  remaining  points  of  intersection 
of  \t  and  2r  with  the  curve.  For  a  triple  point  these  must  coincide. 
Hence 

(T-^):(r-a)(r-7-2):-/3r2+^(7+2)r-(/3-7  +  2/37) 

=  (r— a:):(r  — q:)(t  — 7):  ^Q:r(T  — 7) 


12         I it-a )ul-Ci rcinnscrihed  Triangles  of  Q tun-tic  Curve 

Hence 

(r-/3):-/3T2+/3(7+2)r-(|8-7  +  2/37)  =  (r-a):-ar(r-7) 
and 

(r-^)(r-7-2):-/3T2+^(7  +  2)r-(^-7  +  2/37) 

=  (r  —  a)(r  — 7) :— 0:7(7 —  7) 

That  is, 

(a-/3)r3  +  (/37-a7  +  2/3)T2-(2a^  +  2/37+iS-7)r 

+«(/3-7  +  2i37)=0...:...  (3) 
and 

(a-/3)r3  +  (a-/5)(7+2)r--(/3-7  +  2^7)r+«(/3-7  +  2^7)  =  0...(4) 
Eliminate  r  from  (3)  and  (4) 

£=(/3-l)2(/3-7)(a-|3)=0 

As  previously  shown,  ^  —  y,  a  =  ^,  are  the  conditions  that  a 
branch  of  the  curve  passes  through  one  of  the  cusps.  If  /3=1, 
then  the  quintic  degenerates,  that  is,  there  can  be  no  triple 
point. 

A  second  proof  of  the  non-existence  of  the  triple  point  will  now 
be  given. 

Subject  the  quintic  to  the  quadratic  transformation  1/^  =  —. 

The  curve  becomes  a  quartic  with  the  following  parametric 
equations : 

y.^  =  (t-l)%t-a)(t-y)=t'-{a-\-y-^2)t^+\ay-\-2(a-\-y)-\-l\r- 

—  {a-{-y-\-2ay)t-\-ay 

From  the  five  column  matrix  formed  b}^  the  coefficients  of  the 
above  three  equations,  the  following  ten  determinants  are  derived: 

2Moi2=- (a -18+ 2) 
16Aoi3=(«  +  /3)(a-^  +  2) 

4Aoi4  =  ai3(«-/3  +  !2) 
24Ao23  =-{a  +  ^)  {uy  -^y  +  2a)  -  ( /3  -  7  +  2^7) 

6A024  =^  «[/3(a7  -  /37  +  2a)  +  (/3  -  7  +  2/37)  ] 

4Ao34=-«-(/3-7  +  2i37) 
J)6Ai23  =  7-(«+/3)(a-/3+2)  +  (^  +  7)[2a-  +  (2a+l)(^-7)] 
24Ai24--aiS7'(«-^  +  !2)-a(/3+7)(/3-7  +  '2ai3) 
16Ai34  =  a-(/3  +  7)^/3-7  +  '2«/3) 
24A234=-a''^7(/3-7  +  2i37) 

If  ao-«i^  +  02/'-«3/^+a4^''==0 

and  bo-bit-\-bof'-b,f-'-\-b,f'  =  0 


Oo 

ai 

Oo 

03 

60 

h 

62 

63 

«1 

(12 

as 

04 

61 

62 

63 

64 

Jn-aii<]-('irciniisciibed  TriaiKjlefi  of  QiKtrtic  Ciirre         1>\ 

be  two  quartics  apolar  to  the  three  above  quartics,  then  the 
determinants  formed  from  the  matrix  of  the  coefficients  of  these 
two  latter  equations  are  proportional  to  the  determinants  in  the 
first  matrix;  so  that 

There  is  no  loss  in  generality  in  placing  p=  1. 

The  condition  that  the  quartic  have  a  triple  point  is 


=  ot 


that  is, 

Aoi  .  A34- A02  .  A24  +  A03  .  A23  +  A12  .  Ai4-Ai3"+A23  .  Al2  =  0 

24A2.34 .  24Aoi2-  I6A134 .  16Aoi3+'24Ai24  •  4Aoi4+4Ao34  •  24Ao23 

-(6Ao24)-+4Aoi4.4Ao34  =  0 

Substituting  the  values  of  the  AijkS,  this  becomes 

aH0y(a-l3-\-2)  (l3-y-\-2(3y)  -  (a-]-^)((3-\-y)  (a-/3  +  2)  {^-y-\-2^y) 
+  /3(a-/3  +  2)!^7-(a-/3  +  2)  +  (/3+7)(/3-7  +  2a^)! 
+  (/3-7  +  2/37)|(a  +  i3)  (a7-i87  +  ^2a)+ (/3-7  +  2^7)l 
-\l3(ay-0y-\-ia  +  ){i3-y  +  Wy)\'-\-ct^(a-l^-\-'2)W-y-\-^20y)]  =  O 

Which  reduces  to 

2«^'(/3-l)-(a-/3)(/3-7)(«-7)  =  0 

But  it  has  been  seen  already  that  none  of  the  conditions  here 
obtained  can  be  the  necessary  condition  for  a  triple  point. 

A  third  proof  of  the  non-existence  of  a  triple  point  may  be  stated 
thus: 

If  possible,  let  the  three  cusped  rational  quintic  have  a  triple 
point. 

Subject  the  quintic  to  a  quadratic  transformation. 

Let  the  vertices  of  its  singular  triangle  be  two  of  the  cusps  and 
the  triple  point,  the  cusi)s  being  at  the  vertices  1  and  2,  and  the 
triple  point  at  3. 

The  resulting  curve  is  of  order 

10-3-2-2  =  3,t 

*  W.  F.  Meyer:  "Apolaritat  und  Rationale  Curven,"  Chap,  i,  p.  3. 
t  Ibid.,  Chap,  ii,  p.  184. 

JR.  Sturm:  "Die  Lehre  von  dem  Geometrischen  Verwandtschaften," 
Vol.  iv,  p.  44. 


14         In-(nid-Circiimsc)ihed  Triangles  of  Qitartic  Curve 

having  points  of  tangency  on  the  sides  1'3'  and  2'3'  of  the  singular 
triangle  in  the  transformed  plane;  also  a  branch  of  the  curve  goes 
through  the  vertex  3',  since  there  is  one  extra  intersection  on  the 
side  12  of  the  original  triangle. 

The  original  quintic  had  a  third  cusp,  which  in  the  transforma- 
tion remains  a  cusp.  Hence  the  new  curve  is  a  cuspidal  cubic, 
and  therefore  of  the  third  class.  That  is,  from  a  point  of  the 
curve,  but  one  tangent,  excluding  the  one  at  the  point  itself, 
may  be  drawn. 

But  this  curve  would  be  on  the  vertex  3'  and  have  as  tangents  the 
sides  1'3'  and  2'3',  which  is  clearly  an  impossibility.  i\ccordingly, 
the  three  cusped  rational  quintic  cannot  have  a  triple  point. 


The  Quadratic  Transformation 
t-ai 


If   the  quintic  xi  =  ^^_     '     (i  =  1,   2,   3)    be   subjected   to   the 


quadratic  transformation  Xi  =  -   (i—  1,  2,  3),  the  resulting  curve  is 

t—ai 
or  yi  =  {t-^i)Ht-a2)(t-as) 

y2  =  {t-^2)-{t-a^){t-a{) 
yz  =  {t-^^y{t-a,){t-a.^. 

This  is  a  curve  of  the  fourth  order.  It  passes  through  the 
vertices  of  the  fundamental  triangle  of  the  transformed  plane, 
and  is  at  the  same  time  tangent  to  the  sides.  That  is,  the  funda- 
mental triangle  is  simultaneously  in-and-circumscribed  to  the 
quartic,  its  vertices  being  t=ai,  and  its  points  of  tangency  t  =  ^i. 

It  has  been  seen  that  the  original  quintic  has  three  double 
points;  and,  further,  that  conditions  can  be  imposed  upon  it  so 
that  it  can  acquire  a  fourth  cusp  or  tacnode  without  degenerating. 
By  the  quadratic  transformation,  quintics  of  each  of  these  types 
will  go  into  quartics  having  three  double  points,  a  cusp  or  a  tacnode 
res])ectively. 

The  existence,  therefore,  of  one  triangle  in-and-circumscribed 
to  the  quartic  led  to  the  investigation  of  the  number  of  such 
triangles  in  each  of  these  cases. 

It  was  shown  that  the  rational  quintic  with  three  cusps  could 
not  have  a  triple  point.  Accordingly,  the  rational  quartic  with 
a  triple  point  cannot   have   triangles  in-and-circumscribed;  for, 


hi-and-Circumscrihed  Triangles  of  Quartic  Curve  15 
if  it  were  possible,  by  subjecting  the  quartic  to  the  quadratic 
transformation  i/i  =  -    (with  such  a  triangle  as  reference  triangle), 

it  would  go  into  a  quint ic  with  a  triple  point.  This  has  been  shown 
impossible. 

The  Rational  Quartic 

Consider  now  the.in-and-circumscribed  triangles  of  the  rational 
quartic. 

Let  its  parametric  equations  be : 

Xi  =  aot'^-\-ait^-\-a2t~-{-ast-\-ai 
X2=  biP+bor^+bst 

X,=  CitS-^c2t'-\.c,t 

thus  making  the  vertex  (1,  0,  0)  a  double  point  with  parametric 
values  thereat  /  =  0  and  t=  co. 

Let  (as)  =  0  and  {(Ss)  =  0  be  the  conditions  upon  a  set  of  four 
parameters  that  they  lie  on  a  line. 

Then  aoSo-{-aiSi-{-a2S2-\-oc3Ss-}-aiSi  =  0 

i80'^0  +  l8l5l+|8252+|8353+/S454  =  0, 

where  {a^)ik  is  proportional  to  the  determinant  Aimn  formed 
from  the  matrix  of  the  coefficients  of  the  parametric  equations  of 
the  curve.* 

That  is,       1  q:/5  |  oi  — '2M2,34  =  «4  |  be  \  23  =  fcX,  say, 
\  a^  \  02- lQAui  =  ai\  be  \n  =  kY 
I  a:/3  1 03  —  24Ai24  =  rt4  \  be  \  12  =  kZ 
|«/3  |o4^96Ai23=      \abc\viz  =  P 
1  a/3  1 14  -  '24^023  =  Qo  I  be  \  23  =  X 
la/3  |24-16Aoi3  =  ao  I  6<?  I  i3=F 
I  a/3  I  34  — '2Moi2  =  ao  1  be  \  n  =  Z 
I  a^  I  12  =  1  a/3  I  13  =  I  a/3 1  23  =  4Ao34  =  6A024  =  4Aoi4  =  0 

where  h  =  — -. 
aa 

In  (a5)  =  0  and  (^.'^)=.0  take  si  as  the  symmetric  functions  of 
Xi,  X2,  X,  X,  so  that, 

50=1,     5i  =  Xi+X2+^2X,     .y2=XiX2+2X(Xi+X2)+X2 

53  =  2XiX2X+(Xi,+X2)X2,        6'4-XlX2X2 


*  W.  F.  Meyer:  "Apolaritat  und  Rationale  Curven,"  Chap,  i,  p.  33. 


E  = 
-4 


16         Iii-und-Circumscribed  Triangles  of  Qiiartic  Curve 

Substituting  these  values  for  s„ 

(as)  =  [ao+ai(Xi+X2) +0:2X1X2]  + 2[ai+a2(Xi+X2)+a3XiX2]X 

+  [a2+a3(Xi+X2)+a4XiX2]X2  =  0 
(^5)  =  [i3o+^i(Xi+X2) +/32XiX2]  +  2[/3i+/32(Xi+X2) +/33XiX2]X 

+  [/32  +  /33(Xl+X2) +/34XiX2]X2  =  0 

Eliminate  X  from  these  two  equations : 

0:2 +«3(Xi+X2) +0:4X1X2       ao+o!i(Xi+X2) +02X1X2 1  2 

i32  +  ^3(Xl+X2)  +^4X1X2  i8o  +  ^l(Xl+X2)  +^2X1X2 1 

ao+a3(Xi+X2) +0:4X1X2       ai+a;2(Xi+X2)+03XiX2  I  w 

/32  +  ^3(Xl+X2) +134X1X2  /3i+/32(Xl+X2)+/33XiX2    I   ^ 

I  Q:i+02(Xi+X2) +0:3X1X2        0:0 +o;i(Xi+X2) +02X1X2]   _q 
I  i3i  +/32(Xi  +X2)  +133X1X2       ^o+i3i  (Xi  +X2)  +/32X1X2 1 

Expanding  the  eliminant,  and  substituting  therein  for  the 
(aB)i/s  their  respective  values,  there  results  an  equation  in  Xi 
and  X2,  which  is  the  condition  that  the  join  of  Xi  and  X2  be  a  tangent 
to  the  curve.  By  cyclically  permuting,  the  conditions  that  the 
joins  of  Xo  and  X3,  Xi  and  X3  be  tangents  to  the  curve  can  be  written. 
These  three  equations  are  contained  in  the  following  schema, 
viz.,  equations  (1),  (2),  (3),  page  17. 

From  these  equations  there  are  derived  three  equations  in 
Si,  S2,  S3,  the  symmetric  functions  of  X],  X2,  X3.  They  are  equations 
(4),  (5),  (6)  of  the  schema,  page  17. 

It  may  be  well  to  indicate  here  how  these  equations  have  been 
obtained.  Multiply  equation  (1)  by  X3^  (2)  by  Xl^  (3)  by  X2*. 
Subtract  these  two  by  two,  and  throw  out  the  factors  X3— Xi, 
Xi— X2,  X2— X3.  This  gives  three  new  equations.  Subtracting 
any  two  of  these,  and  throwing  out  one  of  the  above  factors, 
equation  (4)  comes  out.  In  an  analogous  way  by  multiplying  the 
original  equations  by  Xr'^  and  by  Xl^  equations  (5)  and  (6)  are 
derived. 

Equations  (5)  and  (6)  are  linear  in  81  and  S2.  From  these  the 
following  values  of  Si  and  So  in  terms  of  S3  are  derived: 

-YiSs'-\-^VHPX-3kYZ)S,'-\-2k(3PY'-\-iXn^'--^2PXyZ 

-j-u-x7J-8ky'Z'-x'Z)s,'+u-H,Pxz-'-ky/j-'ipy'z 
-5X-'rz+nxy)S3''-\-k'{kZ'-^X'-z-'-2Prz-'-\-ioxy'Z 

+4  ¥*)  S3 + ^Ik'Y'Z  

'^' ~  -X^'~Sz'-{-U-\XY(XZ-^2V')S,^-\-k-'{ IGXY'Z - UXY^Z 

-SX^Z'-UY')Ss^-\-U-n'Z{XZ-^2Y-)S3-k'Y^Z'- 

<iXY'S,'+{X'^U-Y'-U-X-'Z'^-2PX^'-\-10kXY^Z)S3' 
-\-2k{PX~Z-2PXY-'-Xn'-5kXYZJ  +  UkY'Z)Ss' 
-\-2k''i4kY'-Z''-\-fiPY'-4PXYZ-\-iX'Z-SX'-Y^-kXZ')Ss"- 
-\-2kU'^{PZ-3XY)S-i-k*Y' 


S2  = 


-X^Y\S,'-^4k''XY{XZ-'2Y'')S3'+kHl(^XY-Z-4kXY'-Z 
-3X^-Z^-UY')Ss'-^U-n^Z{XZ-2Y~)Ss-k'Y'~Z'- 


pq 
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r< 

IS        In-and-Circumscribed  Triangles  of  Qiiartic  Curve 

These  values  substituted  in  equation  (4)  give  an  equation  of 
the  eleventh  degree  in  S3.  The  constant  term  and  the  coefficients 
of  Ss^^  and  83  vanish  identically,  thus  reducing  the  equation, 
after  dividing  through  by  83^,  to  the  eighth  degree. 

The  coefficients  of  this  octavic  are  such  that  when  F=0,  the 
coefficients  of  83^,  S3'',  and  83,  as  well  as  the  constant  term,  become 
identically  zero,  and  the  equation  then  reduces  to  a  quartic.  This 
condition,  viz.,  F  =  0,  is  the  condition,  as  will  be  seen  later,  that 
the  quartic  have  a  tacnode. 

The  equation  has  eight  possible  solutions,  each  of  which  gives 
a  single  value  for  81  and  82.  There  are  then  eight  sets  of  values 
for  81,  82  and  83,  each  set  leading  to  an  in-and-circumscribed 
triangle,  the  vertices  of  which  are  found  from  the  equation 

X3-8iX2+82X-83  =  0. 

It  is  to  be  noted  that  the  number  of  triangles  given  here  is  the 
same  as  that  found  by  Cayley's  formula  (page  4)  for  this  case. 


The  Quartic  With  a  Tacnode 

Let  the  parametric  equations  of  the  quartic  bs 

a:i  =  ao/^+ai/^+«2^~+a3^+rt4 
X2=  biP:i-b2t^-^bd 

X3=  C2t'^ 

thus  making  the  vertex  (1,  0,  0)  a  tacnodal  point  with  parametric 
values  thereat  /  =  0  and  t—  <^,  and  at  the  same  time  making  .i'3  =  0 
a  tangent  to  the  curve  at  the  same  point. 

Let  {as)  =  0  and  {^s)  =  0  be  the  "Schnittpunktformen,"  that  is, 
the  conditions  that  a  set  of  four  points  he  on  a  hne.  Then  using 
the  notation  of  the  previous  case, 

I  a/S  1  01  — "2^234=  —aib3C2  =  kX,  say, 

I  «/3  I  03  —  2Ml24  =  a4&lC2  =  ^'-^ 

|ai8  I  04-96Ai23=-C2|a6|  n  =  P 
I  a/3  I  14  — 24Ao23=  —aobsC2  =  X 
1  a,S  1  34  -  24/^012  =  (lobiCo  =  Z 

I  ajS|   12=    \al3  I   13=   |a|S|23  =  4Ao34  =  6Ao24  =  -l-Aoi4  =  0 

|a^|o2=l«/3|24=r  =  0 

Substituting  }^  =  0  in  equations  (4),  (5),  (6),  page  17,  the 
following  relations  between  Si,  82,  *S3  result: 

X^S3^S2-{P''-GkXZ)S3-'-2kPZS2S,-\-4kXhSiSs-k'~Z'~S2-  =  0.  . I. 

-X^Sz'~Si-^2PXS3'^-^2kXZS2Ss-4kX\S3-\-k-'ZhSs  =  0.  .  .  .IL 

{X''-^kZ'')S3-2kXZSiS3-2kPZS3-k-'Z'-S2  =  0 III. 

From  II  and  III,  linear  in  Si  and  <S2,  it  follows  that : 

2Xi^kZ''-X'-)Ss'^-\-2kPXZSs  -  kZjU-X'-  -k'-Z')       jy 
'^'~  -SkX-'ZSs  '"     .' 

_-X{^kZ-^-X'-)S,'+2kPXZS,+2kZ{^kX-'-k-Z-) 
^'~  -3k-KXZ-' 

Substituting  these  values  for  .Si  and  .S2  in  I,  there  results  the 
following  equation  in  »S'3: 

4Z2(4^'Z2  -  X^)  (X2  _  A-Z2)  .S3^  -  U'PX-'Z(X''+ 2kZ'')S3' 
-k'~XZ(lGX'-32k^X^Z-'-\-P-'XZ-\-  l6k''Z')Ss- 
-  ^k'PXZ'~{2X--\-kZ'-)S3  -  4/.-''Z2(4Z2  -  kZ-')  (X''  -  kZ-')  =  0. 

19 


2(1         fn-<iii«J-('ir(iiiiiscribe(l  TruuKjlcs  of  Qiitirtic  Curve 

This  equation  factors  into : 

[2Z(.Y+2V  A-Z)(X+  ^JlcZ)S■i~-\-kPXZSs-\-2k'~Z{2X-{-  ^J  kZ) 
(X+VA'Z)] 
[2X(Z-2V  kZ){X-  V  A-Z)83-+A-PA'Z83+2A-2Z(^2A'-  V  kZ) 
{X-^kZ)]^0 VI. 

There  are  four  possible  solutions  for  iS's.  Each  value  gives  a 
single  value  for  81  and  So.  Hence,  there  are  four  sets  of  values 
for  iS,,  each  set  determining  an  in-and-circuinscribed  triangle. 

The  fact  that  equation  VI.  factors  into  two  c^uadratic  factors 
must  be  especially  noticed.  This  indicates  that  the  triangles 
fall  into  two  sets  of  two  each,  and  not  into  one  irreducible  set  of 
four  triangles. 


The  Quartic  with  a  Triple  Point 

Let  (as)  =  0  and  (/3s)  =  0  be  the  "Schnittpunktformen"  apolar 
to  the  three  equations  representing  parametrically  the  rational 
quartic,  viz., 

Then  the  condition  that  the  quartic  have  a  triple  point  is 


ao  tti  0:2  "3 

/3o  ^1  1^2  ^Z 

ai  ai  a3  0:4 

^1  /32  /33  i34 


=  0 


which  reduces  to  |  aj8  1 01  •  |  ccjS  I  34  —  I  a/S  [02  •  |  a^S  1 24  =  0. 

That  is,  I  a/S  1 01  : 1  a^  I  02  =  j  a/3  |  24  :|  «/?  I  34^  which  in  the  notation 
previously  used  is  kX:kY  =  Y :  Z,  or  A'Z=  Y~. 

It  is  possible,  therefore,  to  take  A'=  1,  Y  —  m,  Z  —  m'-,  and 
also  k=\. 

Substituting  these  values  in  equations  (4),  (5),  (6),  page  17, 
they  reduce  to : 

S>mSz^-{P^-^m^)S3''+Si''S2+{Qm-^Pm''-)S2S3+{4>-^mP)SiSz 
-m%2_|_(53_5^^S2)2m3+?n2(S2-Sr)=0 1. 

m2S3'-(2P-rm^)S3'-SiS3-  +  (w^+!2?"P-4)S3+2m2S283 

+  27^i3S3+m2Si  =  0 II. 

m  2S2S3  -  +  2w.SuS3  -  +  ( 1  +  2mP  -  4w ')  S,  2  +  2/»  ''SiSs 

-(2P/?i2-6m)^'3  +  w^.S2  +  w-'  =  0 III. 


In-and-Circuin.scribed  Triangles  of  Qiiartic  Curve        21 
From  equations  II  and  III,  linear  in  Si  and  82,  it  follows  that: 


So 


(Sz  +  m)-' 
■2mhS3^-  (Gm^-'iPmA-DSs'--  (MPm''-^m)S3-{-m^ 


m^(S3-\-m)^ 
Substituting  these  values  in  I,  this  equation  becomes: 

( -  m^-{-'2Pm^-^2m'-P-^m^-\-2Pm  -  1)83^(83  +  ^)  ^  =  0. 
Neither  one  of  the  values  ..S3  =  0,  —  m  gives  proper  triangles. 

The  Quarlic  with  a  Cusp 
Let  the  parametric  equations  be : 

thus  placing  the  cusp  at  the  vertex  (1,  0,  0)  with  /  =  0  thereat,  and 
also  making  the  sides  of  the  triangle  of  reference  tangents  to  the 
curve.  The  sides  13  and  I'-Z  have  points  of  tangency  at  t  =  b 
and  f  —  c  respectively;  while  23  is  a  double  tangent  with  t=  ^a 
at  the  points  of  tangency. 

Let  {as)  =  0  and  {0s)^O  be  the  conditions  that  a  set  of  four 
points  lie  on  a  line,  then, 

\a^\  3io2iAoi2  =  2{2a'-bc){c-b) 
I  a/3 1 23  <>   -UoM=-2aXc-6) 
|a^|i3^   6Ao24  =  a*(c2-fe2) 
I  aj8  I  03  -  2Mi24  =  -  2a  ^bc{c  -  b) 

I  a/3  |oi=  I  a/3  |o2=  1  a/3  [04=  I  a/3  |  — 12=  I  a/3  1 14=  |  a/3  [24  =  0 
In  {as)  =  0,  (j3s)  =  0  let  st  be  the  symmetric  functions  of  Xi,  Xo, 
X,  X.  Substitute  for  Si  their  respective  values.  From  these 
ecjuations  eliminate  X.  The  result  is  the  condition  that  the  join 
of  Xi  and  X2  be  a  tangent  to  the  curve.  Upon  substituting  for 
a^ik  their  respective  values  the  eliminant  reduces  to,  after 
throwing  out  the  linear  factor,  2X1X2— (6+^)(Xi+X2) +  -^<"»  and 
thus  making  it  a  (3  —  3)  correspondence : 

aH6+c)(Xi+X2)='+6a^(Xi+X2)-XiX2-2a^6c(Xi+X2)- 

-8(2a2-6c)Xi%'^-4a^(6+c)(Xi+X2)XiX2-8a%%2=0 

The  linear  factor  thrown  out  is  the  condition  that  the  join  of 
Xi  and  X2  pass  through  the  cusp.     The  condition  imposed  upon 


22    ,     In-and-Ch'Cwmscribed  Triangles  of  Quartic  Curve 

this  join  is  that  the  remaining  two  parameters  shall  be  equal,  a 
condition  which  is  satisfied  if  the  line  passes  through  the  cusp. 

Two  similar  equations,  giving  the  conditions  that  the  joins  of 
Xi  and  X2,  X2  and  X3  be  tangents  to  the  curve,  can  be  derived. 
From  these  three,  then,  the  following  equations  in  Si,  S2,  and  S3, 
the  symmetric  functions  of  Xi,  X2,  and  X3  can  be  derived.     They  are: 

8{2a^-bc)S2Ss-a'ib-\-c){So-S,'-)-\-2a\Ss-2a*bcSi  =  0.  .  .     I. 

(^+^)(S2--SiS3)+6.SVS3+26cS3  =  0 11. 

S3-+(6+c)S.>S3+fecS2-  =  0 III. 

From  III,  S3  =  —  6S2  or  —  082 

c  u  ,;  f       ^  AC   •     TT     •         ^-        (5b-c)S2-\-2b'~c 

bubstitutmg  »S3=  —  002  in  11,  gives  Oi  = 7-^—, — ^ , 

b[b-\-c) 

and  substituting  these  values  in  I,  there  results 

a'b'^l  (b-{-c)(3b^c)  -2c(5b-c)] 


S2  =  0,  or 


a*{5b-c)''-8b\2a--bc){b-\-c) 

Similarly  using  83=  —082,  gives 

(5c -6)  82  + 26c  2 


Si  =  - 


cib+c) 


„      a'cmb-\-c)(b-\-3c)-2h(5c-b)] 

and  02  =  — i7"z rv^i — o  -i/  -»   ■> — 1   wl  , — {■>  or  U. 

a*{oc  —  b)-  —  8c^i'-2a-  —  bc){b-\-c) 

The  value  S2  =  0  evidently  does  not  lead  to  a  solution.  Hence, 
there  are  two  possible  solutions,  one  for  each  value  of  S3. 

It  is  to  be  noted  that  the  number  of  triangles  given  here  is  the 
same  as  that  found  by  Cayley's  formula  (page  4)  for  this  case. 


The  (3  —  3)  Correspondence 

The  in-and-circumscribed  triangles  of  the  cuspidal  quartic 
were  found  by  means  of  a  (3  —  3)  correspondence.  This  corre- 
spondence is  not,  however,  of  the  most  general  kind.  The  most 
general  correspondence  of  this  type  is  that  set  up  by  means  of  the 
lines  drawn  from  a  point  of  a  conic  to  a  line-cubic. 

The  question  then  arises,  what  are  the  conditions  which  the 
line-cubic  must  satisfy,  either  in  itself  or  in  its  relations  to  the 
conic,  in  order  that  the  general  (3  —  3)  correspondence  reduce 
to  the  type  set  up  by  the  method  here  employed? 


In-and-Circmnscrihed  Triangles  of  Quartic  Curve        23 
Take  as  the  conic  the  norm-conic: 

.-To  =  1 

Let  the  hne-cubic  be  (a^)  ^  =  0. 

The  condition  that  the  join  of  two  points  Xi,  Xo,  of  the  conic  be  a 
hne  of  the  cubic  is : 


1         2Xi        Xi 
1  2X2         Xr^ 


._ .  2 

'  ■  ~\2  A2 


where  cri,  a^  are  the  symmetric  functions  of  Xi  and  X2. 

Substitute  these  vahies  in  (a^)^  =  0,  thus  giving  the  equation 
of  the  (3  —  3)  correspondence  between  Xi  and  X2 : 

o  O"!^    I  o  0'lO'2"    I    Q  ■>    I    Q  0"i-0-2  (Ti^ 

Uooocr-i       cim-^  +0222  — «3aooi"^  -roaoo20'2'+ocion    7      ''^'^112-7- 

3 

3ao220'2  —  7Qi220'i  —  3aoi2Ci(r2  =  0 

These  coefficients  being  all  independent,  this  is  the  most  general 
(3  —  3)  correspondence.  The  equation  of  the  (3  —  3)  corre- 
spondence set  up  by  the  rational  cuspidal  quartic  has  but  six 
terms,  lacking  the  constant  term  and  those  in  0-1(72-,  a2,  ai.  In 
order  that  the  general  correspondence  become  of  the  same  type 
as  the  special  one,  then 

OOOI  =  O022  =  ai22  =  fl222  =  0 

The  equation  of  the  line  cubic  then  reduces  to : 

aooo^o^+3aoo2^o"^2+3aoii^o?i-+6aoi2?o|i^2+3aii2^r^2+aiii^i^  =  0 

This  cubic  lacks  the  terms  in  ^2^  and  t2~,  thus  indicating  that  the 
line  0-2  =  0  is  a  double  tangent  of  the  line-cubic,  and  is  at  the  same 
time  a  line  of  the  conic. 

The  results  obtained  indicate  some  of  the  conditions  which 
must  be  imposed  upon  the  line  cubic  so  that  the  general  (3  —  3) 
correspondence  may  reduce  to  the  special  type  here  under  con- 
sideration. It  is  to  be  noted,  at  the  same  time,  that  these  condi- 
tions are  necessary  but  may  not  be  sufficient. 

R.  A.  Roberts,  in  a  paper  entitled,  "On  Polygons  Circumscribed 
about  a  Conic  and  Inscribed  in  a  Cubic,"  proposes  "to  consider 
.  .  .  the  general  problem  of  finding  conies  and  cubics  related 
to  each  other  in  such  a  manner  that  it  may  be  possible  to  circum- 


24         I n-and-Circii inscribed  Triangles  of  Quartic  Curve 

scribe  about  the  conic  an  infinite  number  of  polygons  which  are 
inscribed  in  the  cubic."* 

That  is,  he  takes  the  hnes  of  a  conic  and  the  points  of  a  cubic, 
which  is  the  dual  of  what  has  been  used  above,  and  finds  various 
relations  between  the  two  curves  so  that  the  results  obtained  are 
always  poristic. 

It  has  been  shown  here  that,  if  a  tangent  to  a  point  conic  is  a 
double  line  of  a  line  cubic,  the  solution  cannot  be  poristic.  Dually, 
this  would  say  that  if  a  point  cubic  have  a  double  point  and  this 
double  point  be  on  the  line  conic,  the  solution  cannot  be  poristic. 

The  (4  —  4)  Correspondence 

The  correspondence  set  up  in  the  case  of  the  quartic  with 
three  double  points  is  a  (4  —  4)  correspondence.  As  in  the  pre- 
ceding case,  it  is,  however,  not  of  the  most  general  type.  The 
most  general  (4  —  4)  correspondence  is  set  up  by  means  of  a 
point-conic  and  a  line-quartic.  As  in  the  preceding  instance, 
what  are  the  conditions  which  the  quartic  curve  must  satisfy, 
either  in  itself  or  in  its  relations  to  the  conic,  so  that  the  general 
(4  —  4)  correspondence  reduce  to  the  type  set  up  by  the  method 
employed  here.^ 

Let  the  conic  be  the  norm  conic :  .ro  =  1 ;  Xi  =  2X;  X2  =X-. 

And  let  the  line  quartic  be  (a^)  ^  =  0. 

Then  the  conditions  that  the  joins  of  two  points  Xi  and  X2  of  the 
conic  be  a  line  of  the  quartic  is: 

11         2X1        \r  i 

^'~1  1         2X2         Xs^l 

where  au  ao  are  the  symmetric  functions  of  X]  and  X2. 

Substitute  these  values  in  (a^)^  =  0,  thus  giving  the  equation 
of  the  (4  —  4)  correspondence  between  Xi  and  X2 : 

«oooo0'2  —  4aoooi~^ +4aooo2a'2  +t>flooii — | I<i0ooi2    2    i-oaoo220-2 

—  4aoiii—^  +1200112-^7— —  12aoi22~T- +-4ao2220-2-|-flniiy^  —  -l«iii2-^ 

2 
+  601122— r 4ai222^^   +02222  =  0 

These  coefficients  are  all  independent,  and  so  this  is  the  most 
general  type  of  (4  —  4)  correspondence. 

*  Proceedings  of  the  London  Mathematical  Society,  Vol.  xvii,  p.  158. 


0^2 :  —  T  :  1 


In-and-Circumscrihed  Triangles  of  Qitartic  Curve        25 

The  equation  of  the  (4  —  4)  correspondence  set  up  by  the  rational 
quartic  has  but  twelve  terms,  lacking  those  in  ai^,  ai^a2,  ci^. 
(See  Eqn.  1,  p.  17).  In  order  that  the  general  correspondence 
become  of  the  same  type  as  this  special  one,  then, 

fliiu  =  Qoiii  =  f'lll2  =  0. 

The  equation  of  the  line-quartic  then  reduces  to : 

aoooo^o^+4aoooi^o''^i+4aono2^o^^2+6aooii?o^^i^+ 12a  0012^0 -^i^2+6aoo22^o-^2- 

+  12ao]12^0^l"^2+12aoi22^o|l|2"+4ao222^0^2'''  +  6ail22^1-^2-  +  4ai222^1^2'^ 
+  02222^2'*  =0       ^ 

This  quartic  lacks  the  terms  in  ^i^  and  ^i^,  thus  indicating  that 
the  line  .ri  =  0  is  a  double  line  of  the  quartic.  This  line  is  the  join 
of  the  points  of  the  norm-conic  whose  parameters  are  X  =  0,  X  =  oo , 
which  are  also  the  parameters  at  one  of  the  double  points  of  the 
rational  quartic.  Consequently,  the  lines,  which  are  the  joins  of 
the  points  whose  parameters  are  the  same  as  the  parameters 
at  the  other  two  double  points,  are  also  double  lines  of  the  line- 
quartic.  These  conditions  are  some  of  those  which  the  quartic 
must  satisfy,  but  are  not  necessarily  all. 

In  the  case  of  the  quartic  with  a  tacnode,  the  (4  —  4)  corre- 
spondence has  only  eight  terms  (as  is  readily  seen  by  placing 
}'=0  in  Eqn.  1,  p.  17).  Then  the  general  correspondence,  in 
order  to  be  of  the  same  type,  must  have,  in  addition  to 

fliiii  =  Ooui  =  aiii2  =  0, 

also  CtoOOO  =  QOOOI  =  ai222  =  fl2222  =  0. 

The  line  quartic  then  becomes, 

4aooo2^o'?2+6aooii^o-^r+12aooi2^o"^ie2+6aoo22?o'?2-  +  12aoii2^o^i-^2 

+  12aoi22^0^1^2-+4ao222^0?2^  +  6ail22^r^2^=0. 

The  absence  of  the  terms  in  ^i^  and  ^i'^  indicates  that  the  line 
x^  =  0  is  a  double  line  of  the  quartic;  while  the  absence  of  the  terms 
^o"*  and  ^2^*  indicates  that  the  lines  .ro  =  0  and  .r2  =  0  are  also  lines  of 
the  quartic.  The  absence  of  the  terms  in  ^o^^i  and  ^1^2^  means 
that  the  quartic  passes  through  the  vertices  0  and  2,  and  since 
the  lines  .Vq  =  0  and  .r2  =  0  are  lines  of  the  quartic  these  vertices 
must  be  points  of  tangency. 

It  was  shown,  in  the  case  of  the  quartic  with  three  double- 
points,  that  the  lines  joining  the  points  of  the  conic,  whose  para- 
meters are  the  same  as  the  parameters  at  the  double-points,  are 
lines  of  the  quartic,  so  in  the  case  of  the  tacnodal  quartic,  where 


26        In-tuid-Circumscribed  Triangles  of  Qiiartic  Curve 

two  of  the  three  double-points  come  together  at  the  taenode,  the 
line  joining  the  points  of  the  conic  whose  parameters  are  the 
same  as  those  of  the  remaining  double-point,  is  a  line  of  the 
quartic. 

Reality  of  Solutions 

A  solution  is  to  be  regarded  as  real  if  the  cubic  giving  the 

vertices,  viz., 

t'-S,P+S2t-Ss  =  0, 

has  real  coefficients.  In  all  the  cases  examined,  the  ec^uations 
obtained  enable  Sy  and  »S'2  to  be  expressed  rationally  in  terms  of  S3. 
Every  real  value  of  S3  will,  therefore,  lead  to  real  values  of  .Si  and 
S2,  and,  accordingly,  to  a  real  solution.  The  number  of  real 
solutions  will  therefore  be  simply  the  number  of  real  roots  of  the 
equation  in  S3.  It  may  be  noted  that  the  above  cubic  will  not 
always  lead  to  a  triangle  with  three  real  vertices.  In  the  tacnodal 
case,  the  special  values  given  (p.  30)  yield  four  real  solutions.  Of 
these  four  real  values  of  S3,  only  one  leads  to  a  triangle  with  three 
real  vertices. 


Special  Cases 

The  following  section  is  added  in  order  to  include  a  few  cases 
showing  in-and-circuniscribed  triangles  with  three  real  vertices. 
It  is  to  be  noted  that  in  the  various  cases  it  may  not  be  possible 
to  draw  the  maximum  number  of  real  triangles.  For  the  rational 
quartic,  a  case  was  found  where  three  of  the  possible  eight  tri- 
angles could  be  drawn.  For  the  quartic  with  a  tacnode  only  one 
could  be  found.  However,  in  the  case  of  the  cuspidal  quartic, 
it  was  possible  to  find  two  triangles  with  three  real  vertices  each. 

(a)   The  Rational  Quartic 

It  is  proposed,  here,  to  find  a  rational  quartic  symmetrical  as 
to  the  fundamental  triangle  of  reference. 

Let  the  three  double  points  be  at  the  vertices  of  the  triangle, 

771 "  ~|~  771  "4~  1 

and     have    as    parameters     thereat     (0,      oo),    (l, ), 

771 

(  —  771,  771 --\- 771 -{-!).  These  values  are  determined  as  follows: 
If  the  six  parameters  taken  in  order  of  continuity  along  the  path 
of  the  curve  be  0,  1,  a,  oo,  b,  —  w,  then  a  and  b  can  be  determined 
in  terms  of  m  by  means  of  the  relations, 

(0,  1/a,  oo)  =  (a,  co/b,  -m)  =  {b,  -m/O,  1), 
whence  a  =  m--f/n  +  l  and  b= , 

771 

Then       .,  =  A,^(/-l)(/+^^^^+"^  +  ^) 

771 

.r2  =  ht(t-j-77l)(t-[77l'-\-77l-\-l]) 

.r3=(/-l)(/+m)(^+^^'+J"  +  ^(/-b7z^  +  /»  +  l]) 

It  is  necessary  to  determine  A-i,  ko  so  that  the  curve  have  tri- 
angular symmetry.  This  will  be  the  case  if  the  tangents  at  the 
double  points  be  oppositely  inclined. 

The  tangent  at  0  is,  771-'- V"  =0 

The   tangent  at    00  is,^ 'j^  =0. 

A"!         A*2 

These  are  oppositely  inclined  if  ki-  =  77i~k2-.  Take  ki  =  77ik; 
k2=-k. 

27 


28         In-and-Circumscrihed  Triangles  of  Quartic  Curve 

Substituting  these  values  in  the 'above  equations,  the  equations 

f)2  ~  -4-  fYi  -X-  \ 

of  the  tangents  at  1  and  — are  found  to  be : 

m 

xi  ( 1  +  m)  - + A-.r3  =  0  and  .Ti  ( 1  +  m)  ^ + km  ^-x^  =  0. 

Theseareoppositely  inchnedif  A--  = — .     Take  1c  — — . 

m~  m 

The  parametric  equations  of  the  curve  then  become 

Xi  =  77i{l-\-m)-t{t-l){t-\ — — ) 

m 

It  may  readily  be  shown  that  the  tangents  at  the  remaining 
double  point  are  oppositely  inclined,  and  that,  consequently,  the 
curve  has  triangular  symmetry. 

The  line  .ri  =  .r2  intersects  the  curve,  aside  from  /  =  0,  <»,  in  the 
parameter-pair 

/2-(w2  +  ;7?  +  l)=0 

It  is  required  to  find  what  value  m  must  have  so  that  the  point 

f=V/w^+m+l  may  be  a  vertex  of  an  in-and-circumscribed  tri- 
angle. On  account  of  the  triangular  symmetry,  it  is  clear  that 
.the  other  two  vertices  will  be  the  remaining  intersections  of  the 

tangents  at  t=  —^lm--\-}n-{-l  with  the  curve,  and  that  there  will 
be  three  such  triangles. 

The  equation  of  the  tangent  at  this  point  can  be  derived. 
From  this  the  symmetric  functions  of  the  parameters  at  the  three 
vertices,  in  terms  of  m,  can  be  written  down.  Substituting  these 
in  equation  5,  p.  17,  this  reduces  to 

69m>2_|_3i2mii-f-50877iio+422w9-256-lw«-6238//i7-8736OT'' 

-7160wS-3551//i^-304w3+668w2+396w+64  =  0, 

a  positive  root  of  which  is  m  =  '-2.15. 

The  parametric  equations  of  the  curve  become 

Xi  =  2.15(S.15)H(t-i)(t-^3.615) 

X2=-{3.l5)H{t-\-2.15){t -7.77^5) 

.T3  =  2.15(/-l)(/+2.15)(^+3.615)(^-7.7725).  ' 


The  vertices  of  one  triangle,  given  by  t='^m~-\-m-\-l  and  equation 
of   parameters    giving    the     intersections    of     the    tangent    at 

/  =  -^Jm^+m+l  with  the  curve,  are  f  =  2.788,  l.OJ),  and  6.898. 


In-and-Circumscribed  Triatujlcs  of  Qiiartic  Carve 


29 


On  account  of  the  svmnietrv  of  the  curve,  the  vertices  of  the 
other  two  triangles  may  be  found  in  an  analogous  way.  For  the 
construction  of  these  triangles  see  Figure  1. 


(6)    The  Quarlic  with  a  Tacnode 
Let  the  parametric  equations  be 

.r,  =  /(«-4)(/-|) 
thus  making  the  vertex  (1,  0,  0)  a  tacnodal  point  with  parametric 


30         In-and-Circiiniscribed  Triaiifflest  of  Qiiartic  Curve 

values  thereat  /  =  0,    oo,  and  at  the  same  time  making  Xs  —  0  a 

tangent  to  the  curve  at  the  same  point,  and  also  making  Xi  =  0  a 

4 

double  tangent  with  parametric  values  /  =  3  and  /  =  „  at  the  points 

of  tangency. 

From  the  matrix  of  the  coefficients, 

A  =  — -,     Z^5,     F  =  — -,     A- =10. 


Figure  2. 

Substituting  these  values  in  equation  VI,  p.  20,  there  follows: 

[S32-26S3+I2O]  [-35S3--26S3  +  2XI   1X3X28]  =  0, 

whence  83  =  6,  20,  6.9,  or  -7.65. 

(83  =  6  is  the  only  value  which  leads  to  triangle  with  three  real 
vertices.     Substituting  this  value  in  equations  IV  and  V  (p.  19), 

then 

37 
»Si  =  -r         and         ».S2=11. 
0 


f.  •  ,  .    I. 


In-and-Circintiscribed  Triangles  of  Qiiartic  Curve         31 

The  vertices  of  the  triangle  are  then  given  by  the  roots  of  the 
equation  : 

6 

(i- 1.5)(<-3.535)(/-  1.132)  =  0 

For  the  construction  of  this  triangle,  see  Figure  2. 

(c)    The  Quartic  with  a  Cusp 
The  following  values  for  a,  b,  c,  lead  to  two  solutions,  both  of 
which  give  real  triangles,  as  is  seen  from  the  accompanying  table: 


a 

b 

c 

Si 

So 

S3 

^1 

t2 

^3 

1 

-4 

2 

8.487 

-.177 

-.708 

.283 

8.496 

-.292 

7 

.293 

-.586 

6.94 

.319 

-.264 

For  the  construction  of  these  triangles,  see  Figure  3. 
1 


Figure  3. 


BIOGRAPHICAL  SKETCH 

Joseph  Nelson  Rice  was  born  at  Weymouth,  Nova  Scotia,  on 
the  26th  of  December,  1890.  He  received  his  elementary  and 
high  school  education  at  the  public  school  of  this  town.  In  the 
fall  of  1906  he  entered  St.  Francis  Xavier's  College,  Antigonish, 
N.  S.,  and  w^as  graduated  therefrom  with  the  degree  of  Bachelor 
of  Arts  in  1910.  In  1912  he  received  the  degree  of  Master  of 
Arts.  During  the  years  1910  to  1913  he  was  an  instructor  in  the 
department  of  jNIathematics  at  this  same  college.  In  the  fall  of 
19 13,  he  entered  the  Catholic  University  of  America  as  a  graduate 
student  in  the  department  of  Mathematics.  He  has  followed 
courses  under  Dr.  Landry,  Professor  of  Mathematics;  Dr.  Shea, 
Professor  of  Physics;  and  Mr.  Crook,  Instructor  in  Mechanics. 

He  desires  to  take  this  opportunity  of  expres.sing  his  thanks  to 
Professor  Landry  for  many  valuable  suggestions  offered  during 
the  preparation  of  this  thesis. 


liiiiliii 


UNIVERSITY  OF  CALIFORNIA  UBRARY 


I 


